i'm not too sure where you would set up shop for car camping along harris wash other than the parking @ Tenmile Springs corral (bighorn cyn trailhead) or the Red Breaks canyon trailhead. regardless of location, i would think harris wash would be kinda of a hateful place to car camp, especially in june, for various reasons. if you plan on tackling the dayhikes of harris wash, it'd be best to move camp up into the pinyon/juniper forest at the base of the straight cliffs (fiftymile bench) or behind, up on the Kaiparowits plateau, and pack along a couple cubes of H2O. Plenty of options with solitude. not familiar with Tin Cup Flats. good luck.